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In theory of computation, a branch of theoretical computer science, a deterministic finite automaton (DFA)—also known as deterministic finite state machine—is a finite state machine that accepts/rejects finite strings of symbols and only produces a unique computation (or run) of the automaton for each input string.〔Hopcroft 2001:〕 'Deterministic' refers to the uniqueness of the computation. In search of simplest models to capture the finite state machines, McCulloch and Pitts were among the first researchers to introduce a concept similar to finite automaton in 1943.〔McCulloch and Pitts (1943):〕〔Rabin and Scott (1959):〕 The figure illustrates a deterministic finite automaton using a state diagram. In the automaton, there are three states: S0, S1, and S2 (denoted graphically by circles). The automaton takes a finite sequence of 0s and 1s as input. For each state, there is a transition arrow leading out to a next state for both 0 and 1. Upon reading a symbol, a DFA jumps ''deterministically'' from a state to another by following the transition arrow. For example, if the automaton is currently in state S0 and current input symbol is 1 then it deterministically jumps to state S1. A DFA has a ''start state'' (denoted graphically by an arrow coming in from nowhere) where computations begin, and a set of ''accept states'' (denoted graphically by a double circle) which help define when a computation is successful. A DFA is defined as an abstract mathematical concept, but is often implemented in hardware and software for solving various specific problems. For example, a DFA can model software that decides whether or not online user-input such as email addresses are valid. (see: finite state machine for more practical examples). DFAs recognize exactly the set of regular languages〔 which are, among other things, useful for doing lexical analysis and pattern matching. DFAs can be built from nondeterministic finite automata (NFAs) using the powerset construction method. ==Formal definition== A deterministic finite automaton ''M'' is a 5-tuple, (''Q'', Σ, δ, ''q''''0'', ''F''), consisting of * a finite set of states (''Q'') * a finite set of input symbols called the alphabet (Σ) * a transition function (δ : ''Q'' × Σ → ''Q'') * an initial or start state (''q''''0'' ∈ ''Q'') * a set of accept states (''F'' ⊆ ''Q'') Let ''w = a1a2 ... an'' be a string over the alphabet Σ. The automaton ''M'' accepts the string ''w'' if a sequence of states, ''r0,r1, ..., rn'', exists in ''Q'' with the following conditions: # ''r0'' = ''q''''0'' # ''ri+1'' = δ(''ri'', ''ai+1''), for ''i'' = ''0, ..., n−1'' # ''rn'' ∈ ''F''. In words, the first condition says that the machine starts in the start state ''q''0. The second condition says that given each character of string ''w'', the machine will transition from state to state according to the transition function δ. The last condition says that the machine accepts ''w'' if the last input of ''w'' causes the machine to halt in one of the accepting states. Otherwise, it is said that the automaton ''rejects'' the string. The set of strings that ''M'' accepts is the language ''recognized'' by ''M'' and this language is denoted by ''L(M)''. A deterministic finite automaton without accept states and without a starting state is known as a transition system or semiautomaton. For more comprehensive introduction of the formal definition see automata theory. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Deterministic finite automaton」の詳細全文を読む スポンサード リンク
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